Modern Classical Mechanics

Description

In this modern and distinctive textbook, Helliwell and Sahakian present classical mechanics as a thriving and contemporary field with strong connections to cutting-edge research topics in physics. Each part of the book concludes with a capstone chapter describing various key topics in quantum mechanics, general relativity, and other areas of modern physics, clearly demonstrating how they relate to advanced classical mechanics, and enabling students to appreciate the central importance of classical mechanics within contemporary fields of research. Numerous and detailed examples are interleaved with theoretical content, illustrating abstract concepts more concretely. Extensive problem sets at the end of each chapter further reinforce students' understanding of key concepts, and provide opportunities for assessment or self-testing. A detailed online solutions manual and lecture slides accompany the text for instructors. Often a flexible approach is required when teaching advanced classical mechanics, and, to facilitate this, the authors have outlined several paths instructors and students can follow through the book, depending on background knowledge and the length of their course. Cover Half-title Title page Copyright information Dedication Contents Preface Acknowledgements and Credits Notation and Conventions Useful Relations Part I 1 Newtonian Particle Mechanics 1.1 Inertial Frames and the Galilean Transformation 1.2 Newton's Laws of Motion 1.3 One-Dimensional Motion: Drag Forces 1.4 Oscillation in One-Dimensional Motion 1.5 Resonance 1.6 Motion in Two or Three Dimensions 1.7 Systems of Particles 1.8 Conservation Laws 1.9 Collisions 1.10 Forces of Nature 1.11 Summary Problems 2 Relativity 2.1 Einstein's Postulates and the Lorentz Transformation 2.2 Relativistic Kinematics 2.3 Relativistic Dynamics 2.4 Summary Problems 3 The Variational Principle 3.1 Fermat's Principle 3.2 The Calculus of Variations 3.3 Geodesics 3.4 Brachistochrone 3.5 Several Dependent Variables 3.6 Mechanics from a Variational Principle 3.7 Motion in a Uniform Gravitational Field 3.8 Arbitrary Potential Energies 3.9 Summary Problems 4 Lagrangian Mechanics 4.1 Nonconservative Forces 4.2 Forces of Constraint and Generalized Coordinates 4.3 Hamilton's Principle 4.4 Generalized Momenta and Cyclic Coordinates 4.5 Systems of Particles 4.6 The Hamiltonian 4.7 When is H [neq] E? 4.8 The Moral of Constraints 4.9 Small Oscillations about Equilibrium 4.10 Recap Problems 5 From Classical to Quantum and Back 5.1 Classical Waves 5.2 Two-Slit Experiments and Quantum Mechanics 5.3 Feynman Sum-over-Paths 5.4 Helium Atoms and the Two Slits, Revisited 5.5 The Emergence of the Classical Trajectory 5.6 Why Hamilton's Principle? 5.7 The Jacobi Action 5.8 Summary Problems Part II 6 Constraints and Symmetries 6.1 Contact Forces 6.2 Symmetries and Conservation Laws: A Preview 6.3 Cyclic Coordinates and Generalized Momenta 6.4 A Less Straightforward Example 6.5 Infinitesimal Transformations 6.6 Symmetry 6.7 Noether's Theorem 6.8 Some Comments on Symmetries 6.9 Summary Problems 7 Gravitation 7.1 Central Forces 7.2 The Two-Body Problem 7.3 The Effective Potential Energy 7.4 The Shape of Central-Force Orbits 7.5 Bertrand's Theorem 7.6 Orbital Dynamics 7.7 The Virial Theorem in Astrophysics 7.8 Summary Problems 8 Electromagnetism 8.1 Gravitation Revisited 8.2 The Lorentz Force Law 8.3 The Lagrangian for Electromagnetism 8.4 The Two-Body Problem, Once Again 8.5 Coulomb Scattering 8.6 Motion in a Uniform Magnetic Field 8.7 Relativistic Effects and the Electromagnetic Force 8.8 Summary Problems 9 Accelerating Frames 9.1 Linearly Accelerating Frames 9.2 Rotating Frames 9.3 Pseudoforces in Rotating Frames 9.4 Pseudoforces on Earth 9.5 Spacecraft Rendezvous and Docking 9.6 Summary Problems 10 From Black Holes to Random Forces 10.1 Beyond Newtonian Gravity 10.2 The Schwarzschild Geometry 10.3 Geodesics in the Schwarzschild Spacetime 10.4 The Event Horizon and Black Holes 10.5 Magnetic Gravity 10.6 Gauge Symmetry 10.7 Stochastic Forces 10.8 Summary Problems Part III 11 Hamiltonian Formulation 11.1 Legendre Transformations 11.2 Hamilton's Equations 11.3 Phase Space 11.4 Canonical Transformations 11.5 Poisson Brackets 11.6 Poisson Brackets and Noether's Theorem 11.7 Liouville's Theorem 11.8 Summary Problems 12 Rigid-Body Dynamics 12.1 Rotation About a Fixed Axis 12.2 Euler's Theorem 12.3 Rotation Matrices and the Body Frame 12.4 The Euler Angles 12.5 Infinitesimal Rotations 12.6 Angular Momentum 12.7 Principal Axes 12.8 Torque 12.9 Kinetic Energy 12.10 Potential Energy 12.11 Torque-Free Dynamics Using Euler Angles 12.12 Euler's Equations of Motion and Stability 12.13 Gyroscopes 12.14 Summary Problems 13 Coupled Oscillators 13.1 Linear Systems of Masses and Springs 13.2 More Realistic Bound Systems 13.3 Vibrational Degrees of Freedom 13.4 The Continuum Limit 13.5 Summary Problems 14 Complex Systems 14.1 Integrability 14.2 Conservative Chaos 14.3 Dissipative Chaos 14.4 The Logistic Map 14.5 Perturbation Techniques 14.6 Numerical Techniques 14.7 Summary Problems 15 Seeds of Quantization 15.1 Hamilton–Jacobi Theory 15.2 Hamilton's Characteristic Function 15.3 Action Angle Variables 15.4 Adiabatic Invariants 15.5 Early Quantum Theory 15.6 Optics: From Waves to Rays 15.7 Schrödinger's Wave Mechanics 15.8 Quantum Operators and the Bracket 15.9 A Hint at Quantum Time Evolution 15.10 Summary Problems Appendices A Coordinate Systems B Integral Theorems C Dimensional Reasoning D Fractal Dimension E A Brief on Special Polynomials F Taylor Series Further Reading Index